Solutions to Vector Arithmetic Exercises

1. u + ( v + w ) = OP + ( PQ + QR ) = OR = ( OP + PQ ) + QR ) = ( u + v ) + w .

2. v + ( - v ) = OP + PO = OO = 0 .

3. Both c ( d v ) and ( c d ) v are in the direction of v, and both have cd times v's magnitude. Thus they are the same vector.

4. The magnitude of c 0 is zero, so it must be the zero vector.

5. The magnitude of 0 v is zero, so it must be the zero vector.

6. OP - OQ = OP + ( - OQ ) = OP + QO = QO + OP = QP .

7. In the following window,
   1. the sum of the red and blue vector is magenta,
   2. the red vector minus the blue vector is green,
   3. and three times the blue vector is dark blue.

8. One way to prove this is just by coordinates: c ( v + w ) = c ( ( v₁, v₂, v₃1, w₂, w₃ ) ) = ( c v₁ + c w₁, c v₂ + c w₂, c v₃ + c w₃ ) = c ( v₁, v₂, v₃1, w₂, w₃ ) = c v + cv .

   A more elegant answer uses The Similar Triangles Theorem: let v = OP, w = PQ, c v = OR, and c w = RS . Then tri(O P Q) and tri(O R S) are similar triangles so c segment(OQ) and c ( v + w ) = c v + c w .